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The Selection of ARIMA Models with or without Regressors

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  • Søren Johansen

    (Department of Economics, University of Copenhagen and CREATES, University of Aarhus)

  • Marco Riani

    (Dipartimento di Economia, Universita di Parma)

  • Anthony C. Atkinson

    (Department of Statistics, London School of Economics, UK)

Abstract

We develop a Cp statistic for the selection of regression models with stationary and nonstationary ARIMA error term. We derive the asymptotic theory of the maximum likelihood estimators and show they are consistent and asymptotically Gaussian. We also prove that the distribution of the sum of squares of one step ahead standardized prediction errors, when the parameters are estimated, differs from the chi-squared distribution by a term which tends to infinity at a lower rate than X (2/n). We further prove that, in the prediction error decomposition, the term involving the sum of the variance of one step ahead standardized prediction errors is convergent. Finally, we provide a small simulation study. Empirical comparisons of a consistent version of our Cp statistic with BIC and a generalized RIC show that our statistic has superior performance, particularly for small signal to noise ratios. A new plot of our time series Cp statistic is highly informative about the choice of model. On the way we introduce a new version of AIC for regression models, show that it estimates a Kullback-Leibler distance and provide a version for small samples that is bias corrected. We highlight the connections with standard Mallows Cp.

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Bibliographic Info

Paper provided by University of Copenhagen. Department of Economics in its series Discussion Papers with number 12-17.

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Length: 30 pages
Date of creation: 08 Nov 2012
Date of revision:
Handle: RePEc:kud:kuiedp:1217

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Related research

Keywords: AIC; ARMA models; bias correction; BIC; Cp plot; generalized RIC; Kalman filter; Kullback-Leibler distance; state-space formulation;

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  1. Peide Shi & Chih-Ling Tsai, 2004. "A Joint Regression Variable and Autoregressive Order Selection Criterion," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(6), pages 923-941, November.
  2. Hansheng Wang & Guodong Li & Chih-Ling Tsai, 2007. "Regression coefficient and autoregressive order shrinkage and selection via the lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(1), pages 63-78.
  3. Mike K. P. So & Cathy W. S. Chen & Feng-Chi Liu, 2006. "Best subset selection of autoregressive models with exogenous variables and generalized autoregressive conditional heteroscedasticity errors," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 55(2), pages 201-224.
  4. Claeskens,Gerda & Hjort,Nils Lid, 2008. "Model Selection and Model Averaging," Cambridge Books, Cambridge University Press, number 9780521852258, April.
  5. Marc K. Francke & Siem Jan Koopman & Aart de Vos, 2008. "Likelihood Functions for State Space Models with Diffuse Initial Conditions," Tinbergen Institute Discussion Papers 08-040/4, Tinbergen Institute.
  6. Qiwei Yao & Peter J. Brockwell, 2006. "Gaussian Maximum Likelihood Estimation For ARMA Models. I. Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(6), pages 857-875, November.
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